C CS Phys 191 Entangled Spins Intro to Atomic Qubits Spring 2005 3 03 05 Lecture 14 1 Entanglement and Spin So far we ve talked about 1 qubit operations e g 0 1 But what about entanglement What about when there are more than one qubits Question How do we physically create an entangled state of 2 spins Answer Must have an interaction between them i e two particle Hamiltonian How do we create such an interaction and how does it lead to entanglement Let s start with two physical qubits say electrons Claim The ground state of this system is an entangled state Namely 21 0 1 1 2 1 1 0 2 a Bell state How do we show this It s the same old quantum story solving the Schr equation So what is H We must figure out how these electrons interact with each other What effect could one electron have on the other electron and vice versa Well we know that an electron has a magnetic dipole moment that is related to its spin Magnetic dipole moments come up classically when you have current loops so conceptually we can think of electrons as little current loops that generate dipolar magnetic fields But if one electron generates a magnetic field then the other electron can feel that electric field If you put two electrons close enough together then we imagine that the two generated fields from electron 1 2 will be felt by electron 2 1 What does this look like mathematically i e what is the Hamiltonian Well we know me S and H B But we know that the magnetic field produced by a magnetic moment is proportional to that C CS Phys 191 Spring 2005 Lecture 14 1 vector so we can say B1 S1 or B1 A S1 where A 0 So e H S2 A S1 m 1 or equivalently H C S2 S1 2 where C 0 This is our electron interaction Hamiltonian You might wonder what C is C should be a strong function of the relative position but we don t want to worry about that now You can just imagine that C is determined through experiment but is equal to some value that is yet to be determined The important part is the S1 S2 term So what is the ground state of this Hamiltonian Let s use a little trick whenever you hear trick in regards to spins and angular momentum it s time to bust out the raising lowering operators Consider STotal S1 S2 a new operator that should represent the total spin of the two electron system Let s look at some features of this new operator ST ST S T2 S1 S2 S1 S2 S 12 S 22 2 S2 S1 3 We see that the dot product S2 S1 has appeared Let s solve for this quantity S2 S1 1 S T2 S 12 S 22 2 4 and therefore our interaction Hamiltonian can be expressed as H C 2 S T S 12 S 22 2 5 state will be whatever state minimizes the expectation value of this operator recall E So the ground H Note that no matter what is S 12 h 2 21 21 1 43 h 2 The same goes for S 22 so we can replace both S 12 and S 22 with 43 h 2 We see that the Hamiltonian can be rewritten as H DS T2 F 6 where D and F are constants greater zero So what state has the smallest S T2 The best we than could hope for is such that S T2 0 C CS Phys 191 Spring 2005 Lecture 14 2 It is left to a homework problem to show that the following state o 1 0 1 1 2 1 1 0 2 2 7 is an eigenstate of S T2 with eigenvalue 0 We conclude from this that we can experimentally create a Bell state by putting 2 spins next to each other and then providing a perturbation such that the fall into the ground state 2 Interaction Hamiltonians in General Electron spins are hardly the only quantum system that can be entangled so we might want to talk about general entanglement through interaction Hamiltonians In the most general sense we can talk about two quantum numbers x1 and x2 i e quantized physical observables that yield the following Hamiltonian H x1 x2 H 1 x1 H 2 x2 H x1 x2 8 where H x1 x2 6 H 1 x1 H 2 x2 If this is the case then resultant energy eigenstates will not in general be product states x1 x2 6 1 x1 2 x2 9 This is the definition of entanglement Therefore in our quest for entanglement we need to focus our attention on interaction Hamiltonians In the case of spins H CS 1 S 2 is such a Hamiltonian and we will explore others in coming lectures 3 Introduction to Atomic Qubits Whenever we think of qubits we will try to draw analogies to the spin 1 2 system but let s now consider another very useful quantum system Atoms What is an atom How can it be thought of as a qubit Quick answer An atom is a tiny box that holds electrons in discrete energy levels If I can focus on one particular electron that can hop between 2 different states then that is a qubit Question How do we measure and manipulate this qubit Equivalently how do we control the state of the valence electron Answer Apply a Hamiltonian Apply an external field that leads to some H to quote Dr Evil this is usually done with a laser Once we ve created the desired Hamiltonian as usual we solve the Schr Equation is this sounding like a broken record yet The Hamiltonian drives development of the system through the time develop the time ment operator e iH t h and we can change C CS Phys 191 Spring 2005 Lecture 14 3 Note if we keep our focus on only one electron hopping between two states then this problem is identical to the spin 1 2 problem We have to relabel some quantities but the physics and the ultimately results are the same Our atomic state specified by the state of the valence electron 0 1 can be thought of as a vector on the Bloch sphere even though it s not spin 0 and 1 actually are and how do we compute To move forward however we must have a better idea of H and its action on 0 and 1 are atomic energy levels for a single atom Equivalently we could say that 0 and 1 are the energy eigenstates for an electron orbiting a nucleus H o 0 Eo 0 10 H o 1 E1 1 So what is the Hamiltonian H o describing this We can focus on hydrogen because it s conceptually simple Practically experimentalists don t usually work with hydrogen because it s pretty tricky to deal with A large chunk of modern atomic physics esp quantum computing research is done with hydrogenic …